Colloquium

The Battle for Cantorian Set Theory

10/26/2016 - 4:15pm
Speaker: 
Joseph Dauben (CUNY)
Abstract: 

The substance of Georg Cantor’s revolutionary mathematics of the infinite is well-known: in developing what he called the arithmetic of transfinite numbers, he gave mathematical content to the idea of actual infinity. In so doing he laid the groundwork for abstract set theory and made significant contributions to the foundations of the calculus and to the analysis of the continuum of real numbers. Cantor’s most remarkable achievement was to show, in a mathematically rigorous way, that the concept of infinity is not an undifferentiated one. Not all infinite sets are the same size, and consequently, infinite sets can be compared with one another. But so shocking and counter-intuitive were Cantor’s ideas at first that the eminent French mathematician, Henri Poincaré, once characterized Cantor’s theory of transfinite numbers as a “disease” from which he was certain mathematics would one day be cured. Leopold Kronecker, one of Cantor’s teachers and among the most prominent members of the German mathematics establishment, even attacked Cantor personally, and in the words of Arthur Schoenflies, called him a “scientific charlatan,” a “renegade” and a “corrupter of youth.” It is also well-known that Cantor suffered throughout his life from a series of “nervous breakdowns” which became increasingly frequent and debilitating as he got older. Some have linked this to his dangerous flirtations with the infinite, and Cantor has often been portrayed as the hapless victim of the infinite, due to his increasingly long periods of mental breakdown that began in the 1880s, all of which were exacerbated by the persecutions of his contemporaries. But such accounts distort the truth by trivializing the genuine intellectual concerns that motivated some of the most thoughtful contemporary opposition to Cantor’s theory. They also fail to credit the power and scope of the defense Cantor offered for his ideas in the battle to win acceptance for transfinite set theory.

Where: 
Kravis Lower Court, Claremont McKenna College

Patterns of Synchrony: From Animal Gaits to Binocular Rivalry

04/20/2016 - 4:15pm
04/20/2016 - 5:15pm
Speaker: 
Martin Golubitsky (OSU)
Abstract: 

This talk will review previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.

 

Where: 
Argue Auditorium, Millikan, Pomona College

Approximation theory perspective on data processing

04/06/2016 - 4:15pm
04/06/2016 - 5:15pm
Speaker: 
Hrushikesh Mhaskar (CGU/Caltech)
Abstract: 

We argue that data based classification and regression problems can be viewed profitably as problems of function approximation. We describe in detail a case study in the theory of function approximation in case of approximation of continuous 2*pi–periodic functions by trigonometric polynomials. In particular, we will discuss the equivalence theorems, and some relatively new research regarding local approximation based on Fourier information. Several extensions to other settings are mentioned, and a few applications and potential applications are discussed.

Where: 
Argue Auditorium, Millikan, Pomona College

Fixed Point Free Permutations

02/17/2016 - 4:15pm
Speaker: 
Robert Guralnick (USC)
Abstract: 

A permutation g on a set X is said to be fixed point free (or a derangement) if g fixes no points.   If G is a finite group acting transitively on a set of cardinality greater than 1, then an elementary 1872 theorem of Jordan says that G contains derangements.  We will discuss generalizations of this result and applications to finite fields and number theory.

 

Where: 
Argue Auditorium, Millikan, Pomona College

Complicated triangulations and breast cancer research

02/10/2016 - 4:15pm
02/10/2016 - 5:15pm
Speaker: 
Mimi Tsuruga (UC Davis)
Abstract: 

Topology is considered to be one of the more pure (and perhaps? less useful) subjects in mathematics. However, the last two decades have seen a surge in research in applied topology, where topological tools have been employed to solve problems in image processing, dynamical systems, neuroscience, and virus evolution. In this talk, we will give a very basic introduction to a few topological tools and how some of them can be used to analyze genomic data of breast cancer.

Where: 
Argue Auditorium, Millikan, Pomona College

Dynamical Systems from a Number Theory Perspective

01/18/2016 - 3:45pm
01/18/2016 - 4:45pm
Speaker: 
Joe Silverman
Abstract: 

Dynamics is the study of iteration.  Classically one takes a rational map f(z)=F(z)/G(z) and attempts to describe the behavior of points under interation f^n(z) = f(f(f(...f(f(z))...))). The *f-orbit* of a point b is the set of images of b under the repeated iteration of f, Orbit of b = { b, f(b), f^2(b), f^3(b), ... }. The points with finite orbit are called *preperiodic points*. They play a particularly important role in the dynamics of f. For a number theorist, it is natural to take F(z) and G(z) to be polynomials with integer coefficients and to study the orbits of rational numbers b. In this talk I will survey some of the known results and some of the outstanding conjectures related to this number-theoretic view of dynamics.

Typical problems include: 

(1) How many preperiodic points can be rational numbers?

(2) For which rational maps f can the orbit of a rational number b contain infinitely many integers?

 

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Squaring the Circle and Other Impossible Dreams: Periods and Transcendence

04/13/2016 - 4:15pm
04/13/2016 - 5:15pm
Speaker: 
Paula Tretkoff (TAMU)
Abstract: 

The phrase “squaring the circle” is a recognized idiom of the English language that has come to mean “to solve an unusually hard problem”. This idiomatic meaning is very much reflected in the history of the ancient mathematical problem that gave rise to it: roughly speaking, the problem asks if it is possible to construct a square with the same area as the unit circle using compass and straightedge. The impossibility of squaring the circle is equivalent to the transcendence of the number π: that is, to showing there is no non-zero polynomial P(x) with rational coefficients such that P(π) = 0. Proving the transcendence of π was indeed unusually hard and was only achieved in 1882 by Lindemann, following a method due to Hermite, who proved the transcendence of e in 1873. The transcendence of π is equivalent to that of 2πi, a period of the function exp(z) in that exp(z + 2πi) = exp(z). Moreover e is the special value exp(1). The work of Hermite and Lindemann opened the way to a grand modern theory for proving the transcendence of numbers related to periods and values at algebraic points of other classical functions, like elliptic and, more generally, abelian functions, as well as hypergeometric functions. In this talk, we focus on periods in transcendental number theory, the evolution of results, for example, Hilbert’s seventh problem, the breakthroughs of Baker (Fields Medal 1970) and W¨ustholz, as well as some on some of my work, in part joint with M.D. Tretkoff. We also discuss challenges for the future. The talk presumes the intrigue of impossible dreams but does not assume any background in transcendence.

Where: 
Argue Auditorium, Millikan, Pomona College

Turan's Problem

03/23/2016 - 4:15pm
03/23/2016 - 5:15pm
Speaker: 
Annie Raymond (University of Washington)
Abstract: 

What is the maximum number of edges in a graph on n vertices without triangles? Mantel's answer in 1907 that at most half of the edges can be present started a new field: extremal combinatorics. More generally, what is the maximum number of edges in a n-vertex graph that does not contain any subgraph isomorphic to H? What about if you consider hypergraphs instead of graphs? We will explore different strategies to attack such problems, calling upon combinatorics, integer programming, semidefinite programming and flag algebras. We will conclude with some recent work where we embed the flag algebra techniques in more standard methods.

 

This is joint work with Mohit Singh and Rekha Thomas.

Where: 
Argue Auditorium, Millikan, Pomona College

Mathematical modeling of blood clot formation under flow and microorganism motility in heterogeneous fluids

03/09/2016 - 4:15pm
03/09/2016 - 5:15pm
Speaker: 
Karin Leiderman (UC Merced)
Abstract: 

Many biological fluid environments are inherently heterogeneous, comprised of macromolecules, polymers networks, and/or cells. These environments can be thought of, in some sense, as fluids interacting with biological porous materials. In this talk I will describe mathematical models of two such situations: dynamic formation of blood clots under flow and motility of microorganisms through the reproductive tract. The model of clot formation was developed to better understand the interplay between flow-mediated transport of clotting proteins and cells, blood cell deposition at an injury site, complex clotting biochemistry, and the overall effect of these on clot formation. A derivation of hindrance coefficients for the advection and diffusion of clotting proteins will be presented. Results show that the effects of such hindrance are profound and suggest a possible physical mechanism for limiting clot growth. Next I will describe regularized fundamental solutions to the governing equations of flow through porous material and the corresponding numerical method used to model a swimming microorganism. Results show that for certain ranges of porosity of the material through which they swim, a single swimmer's propulsion is faster and more efficient, and two swimmers are attracted toward one another.

Where: 
Argue Auditorium, Millikan, Pomona College

A Game Theoretic Approach to Pediatric Vaccine Pricing

03/02/2016 - 4:15pm
03/02/2016 - 5:15pm
Speaker: 
Banafsheh Behzad
Abstract: 

Pricing strategies in the United States pediatric vaccines market are studied using a Bertrand-Edgeworth-Chamberlin price game. The game analyzes the competition between asymmetric manufacturers with limited production capacities and linear demand, producing differentiated products. The model completely characterizes the unique pure strategy equilibrium in the Bertrand-Edgeworth-Chamberlin competition in an oligopoly setting. In addition, the complete characterization of mixed strategy equilibrium is provided for a duopoly setting. The results indicate that the pure strategy equilibrium exists if the production capacity of manufacturers is at their extreme. For the capacity regions where no pure strategy equilibrium exists, there exists a mixed strategy equilibrium. In a duopoly setting, the distribution functions of the mixed strategy equilibrium for both manufacturers are provided. The proposed game is applied to the United States pediatric vaccine market, in which a small number of asymmetric vaccine manufacturers produce differentiated vaccines. The sources of differentiation in the competing vaccines are the number of medical adverse events, the number of different antigens, and special advantages of those vaccines. The results indicate that the public sector prices of the vaccines are higher than the vaccine equilibrium prices. Furthermore, the situation when shortages of certain pediatric vaccines occur is studied. Market demand and degree of product differentiation are shown as two key factors in computing the equilibrium prices of the vaccines.

 

Where: 
Argue Auditorium, Millikan, Pomona College
Add to calendar

Claremont Graduate University | Claremont McKenna | Harvey Mudd | Pitzer | Pomona | Scripps
Proudly Serving Math Community at the Claremont Colleges Since 2007
Copyright © 2018 Claremont Center for the Mathematical Sciences

Syndicate content